If roots of equation $\displaystyle x^2 + x + r = 0$ are $\displaystyle \alpha$ and $\displaystyle \beta$ and $\displaystyle \alpha^3 + \beta^3 = -6$. Find the value of 'r'

If $\displaystyle \frac{3}{x+y} + \frac{2}{x-y} = -1$ and $\displaystyle \frac{1}{x+y} - \frac{1}{x-y} = \frac{4}{3}$ then $\displaystyle (x, y)$ is:

$\displaystyle \frac{2x+5}{10} + \frac{3x+10}{15} = 5$, find $\displaystyle x$

Find value of $\displaystyle x^2 - 10x + 1$ if $\displaystyle x = \frac{1}{5-2\sqrt{6}}$

The cost of $\displaystyle 2$ oranges and $\displaystyle 3$ apples is $\displaystyle 28$. If the cost of an apple is doubled then the cost of $\displaystyle 3$ oranges and $\displaystyle 5$ apples is $\displaystyle 75$. The original cost of $\displaystyle 7$ oranges and $\displaystyle 4$ apples (in Rs) is:

In a multiple choice question paper consisting of $\displaystyle 100$ questions of $\displaystyle 1$ mark each, a candidate gets $\displaystyle 60\%$ marks. If the candidate attempted all questions and there was a penalty of $\displaystyle 0.25$ marks for wrong answers is:

The values of $\displaystyle x$ and $\displaystyle y$ satisfying the equations $\displaystyle \frac{3}{x+y} + \frac{2}{x-y} = 3$ and $\displaystyle \frac{2}{x+y} + \frac{3}{x-y} = \frac{23}{3}$ given by